Finite Group Databasebeta

← #37547706

Subgroups — #37547706

Order 23168.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26042005C211yesmaximal, characteristic, complemented
2#26041999C213yesmaximal, complemented
3#26041984C213yesmaximal, complemented
4#26041979C213yesmaximal, complemented
5#26041964C213yesmaximal, complemented
6Q8×D724C211yesmaximal, characteristic, complemented
7#26041936C211yesmaximal, characteristic, complemented
8#26041936C211yesmaximal, characteristic, complemented
9#26041894C213yesmaximal, complemented
10#26041891C213yesmaximal, complemented
11#26041811C213yesmaximal, complemented
12#26041807C213yesmaximal, complemented
13#26041755C211yesmaximal, characteristic, complemented
14C4×Q8×D181C211yesmaximal, characteristic, complemented
15#520×C181C211yesmaximal, characteristic, complemented
16C2×Q8×D181C2211yescharacteristic
17C2×Q8×D181C2211yescharacteristic
18#19153816C2213yes—
19#19153815C2213yes—
20#19153811C2213yes—
21#19153809C2216yes—
22#19153809C2216yes—
23#19153809C2213yes—
24#19153808C2213yes—
25#19153807C2213yes—
26#19153806C2213yes—
27#19153806C2216yes—
28#19153803C2213yes—
29#19153803C2213yes—
30#19153802C2213yes—
31#19153801C2211yescharacteristic
32#19153799C2216yes—
33#19153798C2213yes—
34#19153796C2213yes—
35#19153796C2213yes—
36#19153795C2213yes—
37#19153795C2213yes—
38#19153795C2216yes—
39#19153794C2213yes—
40#19153789C2213yes—
41#19153787C2213yes—
42#19153787C2213yes—
43Q8×Dic181C2211yescharacteristic
44Q8×Dic181C2211yescharacteristic
45C4×Dic362C2213yes—
46C4×Dic362C2213yes—
47C4×Dic362C2213yes—
48#19153781C2213yes—
49#19153781C2213yes—
50C4×D724C2213yes—
51C4×C22⋊D181C2213yes—
52C4×C22⋊D181C2213yes—
53C4×C22⋊D181C2216yes—
54#19153775C2216yes—
55#19153774C2213yes—
56#19153773C2213yes—
57C4⋊C4×D181C2213yes—
58C42×D181C2213yes—
59C2×#41×C181C2211yescharacteristic
60C4×Q8×C181C2211yescharacteristic
61C4×Q8×C181C2211yescharacteristic
62C4×D4×C181C2213yes—
63C4×D4×C181C2213yes—
64#117×C181C2213yes—
65C2×C4⋊C4×C181C2213yes—
66C2×Dic362C2313yes—
67C2×Dic362C2313yes—
68C2×Dic362C2313yes—
69Q8×D181—24no—
70Q8×D181—24no—
71C2×D724C2311yescharacteristic
72C2×C22⋊D181C2313yes—
73C2×C22⋊D181C2313yes—
74C2×C4×D181C2311yescharacteristic
75C2×C4×D181C2313yes—
76C2×C4×D181C2313yes—
77C2×C4×D181C2313yes—
78C4⋊Dic181C2313yes—
79C4⋊Dic181C2313yes—
80C4⋊Dic181C2313yes—
81C4⋊Dic181C2316yes—
82C4⋊Dic181C2313yes—
83#15668428C2316yes—
84#15668428C2313yes—
85#15668428C2313yes—
86#15668428C2313yes—
87#15668428C2313yes—
88#15668428C2316yes—
89#15668428C2311yescharacteristic
90#15668428C2313yes—
91C22⋊Dic181C2313yes—
92C22⋊Dic181C2316yes—
93C22⋊Dic181C2316yes—
94C22⋊Dic181C2313yes—
95#15668426C2311yescharacteristic
96#15668426C2313yes—
97#15668426C2311yescharacteristic
98#15668426C2311yescharacteristic
99#15668426C2313yes—
100#15668426C2313yes—
101C4×Dic181C2313yes—
102C4×Dic181C2313yes—
103C4×Dic181C2313yes—
104C4×Dic181C2313yes—
105C4×Dic181C2316yes—
106#41×C181—24no—
107C2×Q8×C181C2311yescharacteristic
108C2×D4×C181C2313yes—
109C22×C4×C181C2313yes—
110C22×C4×C181C2316yes—
111C4⋊C4×C181C2313yes—
112C4⋊C4×C181C2313yes—
113C4⋊C4×C181C2313yes—
114C4⋊C4×C181C2311yescharacteristic
115C22⋊C4×C181C2313yes—
116C22⋊C4×C181C2313yes—
117C42×C181C2313yes—
118C42×C181C2313yes—
119C22×D181C2411yescharacteristic
120C22×D181C2411yescharacteristic
121Dic362—26no—
122Dic362—26no—
123Dic362—43no—
124D724—22no—
125C22⋊D181—26no—
126C22⋊D181—43no—
127C2×Dic181C2411yescharacteristic
128C2×Dic181C2411yescharacteristic
129C2×Dic181C2413yes—
130C2×Dic181C2413yes—
131C2×Dic181C2413yes—
132C2×Dic181C2413yes—
133C4×D181—22no—
134C4×D181—26no—
135C4×D181—26no—
136C4×D181—43no—
137C23×C181C2413yes—
138Q8×C181#4114yes—
139D4×C181—26no—
140C2×C4×C181—26no—
141C2×C4×C181—46no—
142C2×C4×C181C2411yescharacteristic
143C2×C4×C181C2411yescharacteristic
144C2×C4×C181C2413yes—
145C2×C4×C181C2413yes—
146C2×C4×C181C2411yescharacteristic
147C2×C4×C181C2413yes—
148C22×C181—23no—
149C22×C181—23no—
150C22×C181—23no—
151C22×C181C2511yescharacteristic
152C2×D181—21no—
153C2×D181—21no—
154C2×D181—21no—
155C2×D181—21no—
156C2×D181—21no—
157C2×D1812−1+412yes—
158C4×C181—21no—
159C4×C181—21no—
160C4×C181—21no—
161C4×C181—23no—
162C4×C181—23no—
163C4×C181C2×#4116yes—
164Dic181—23no—
165Dic181—23no—
166Dic181—21no—
167Dic181—23no—
168Dic181—23no—
169Dic181—21no—
170C2×C181—43no—
171C2×C181C22×#4111yescharacteristic
172C2×C181C2×2−1+411yescharacteristic
173C2×C181#58511yescharacteristic
174D181—22no—
175D181—41no—
176C181#351611yescharacteristic, complemented, Sylow
177#3516—1811nomaximal, Sylow
178#557—1811no—
179#556—1813no—
180#554—1813no—
181#553—1813no—
182#552—1813no—
183D4×Q8—1811no—
184#548—1811no—
185#548—1811no—
186#542—1813no—
187#541—1813no—
188#529—1813no—
189#529—1813no—
190#520D18111yescharacteristic, complemented
191#519—1811no—
192C2×C4×Q8—1811no—
193C2×#41C2×D18111yescharacteristic
194C22×Q8—1811no—
195C22×Q8—1811no—
196C4⋊Q8—1813no—
197C4⋊Q8—1813no—
198#126—1813no—
199#126—1816no—
200#126—1813no—
201#126—1816no—
202#125—1813no—
203#125—1813no—
204#125—1816no—
205#125—1813no—
206#124—1813no—
207#124—1813no—
208#124—1813no—
209#123—1816no—
210#123—1811no—
211#123—1813no—
212C22⋊Q8—1813no—
213C22⋊Q8—1813no—
214C22⋊Q8—1816no—
215C22⋊Q8—1813no—
216C22⋊Q8—1813no—
217C22⋊Q8—1813no—
218#121—1813no—
219#121—1813no—
220#121—1813no—
221C4×Q8—1811no—
222C4×Q8—1811no—
223C4×Q8—1813no—
224C4×Q8—1813no—
225C4×Q8—1813no—
226C4×Q8—1813no—
227C4×Q8—1813no—
228C4×Q8C2×D18111yescharacteristic
229C4×Q8C2×D18111yescharacteristic
230C4×D4—1813no—
231C4×D4—1813no—
232C4×D4—1816no—
233C4×D4—1813no—
234C4×D4C2×D18113yes—
235C4×D4C2×D18113yes—
236#117—1816no—
237#117—1813no—
238#117—1813no—
239#117C2×D18113yes—
240C2×C4⋊C4—1813no—
241C2×C4⋊C4C2×D18113yes—
242C2×C42—1813no—
243#41—24no—
244C2×Q8—3624no—
245C2×Q8—3624no—
246C2×Q8—1813no—
247C2×Q8—1813no—
248C2×Q8—1813no—
249C2×Q8C22×D18111yescharacteristic
250C2×D4—1813no—
251C2×D4—1813no—
252C2×D4—1811no—
253C2×D4C22×D18113yes—
254C22×C4—1811no—
255C22×C4—1813no—
256C22×C4—1813no—
257C22×C4—1813no—
258C22×C4C22×D18113yes—
259C22×C4C22×D18116yes—
260C4⋊C4—1813no—
261C4⋊C4—1816no—
262C4⋊C4—1813no—
263C4⋊C4—1813no—
264C4⋊C4—1813no—
265C4⋊C4—1813no—
266C4⋊C4—1816no—
267C4⋊C4—1813no—
268C4⋊C4—1813no—
269C4⋊C4—1816no—
270C4⋊C4—1813no—
271C4⋊C4—1811no—
272C4⋊C4—1813no—
273C4⋊C4C22×D18113yes—
274C4⋊C4C22×D18113yes—
275C4⋊C4C22×D18113yes—
276C4⋊C4C22×D18111yescharacteristic
277C22⋊C4—1813no—
278C22⋊C4—1816no—
279C22⋊C4—1816no—
280C22⋊C4—1813no—
281C22⋊C4—1813no—
282C22⋊C4—1811no—
283C22⋊C4—1811no—
284C22⋊C4—1813no—
285C22⋊C4—1811no—
286C22⋊C4—1813no—
287C22⋊C4C22×D18113yes—
288C22⋊C4C22×D18113yes—
289C42—1813no—
290C42—1813no—
291C42—1813no—
292C42—1816no—
293C42—1813no—
294C42C22×D18113yes—
295C42C22×D18113yes—
296C23—1811no—
297C23—1811no—
298C23C23×D18113yes—
299Q8—3626no—
300Q8—3626no—
301Q8—7243no—
302Q8#1566844914yes—
303D4—26no—
304D4—3622no—
305D4—3626no—
306D4—7243no—
307C2×C4—26no—
308C2×C4—46no—
309C2×C4—3622no—
310C2×C4—3626no—
311C2×C4—3626no—
312C2×C4—1811no—
313C2×C4—1811no—
314C2×C4—1813no—
315C2×C4—1813no—
316C2×C4—1813no—
317C2×C4—1813no—
318C2×C4—7243no—
319C2×C4C23×D18111yescharacteristic
320C2×C4C23×D18113yes—
321C2×C4C23×D18111yescharacteristic
322C2×C4C23×D18113yes—
323C2×C4C23×D18111yescharacteristic
324C2×C4C23×D18113yes—
325C22—23no—
326C22—23no—
327C22—23no—
328C22—3621no—
329C22—3621no—
330C22—3621no—
331C22—3621no—
332C22—3621no—
333C22—1812no—
334C22C24×D18111yescharacteristic
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—3623no—
341C4—3621no—
342C4—3623no—
343C4—3623no—
344C4—3621no—
345C4—3623no—
346C4C2×#1566844916yes—
347C2—43no—
348C2—3622no—
349C2—7241no—
350C2C22×#1566844911yescharacteristic
351C22−1+4×D18111yescharacteristic
352C2#2604211411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D181
commutator, socle, cosocleC22×C181C25
Fitting#520×C181C2
hypercenter#520D181
hypocenterC181#3516
LCSG ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ #520

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #26042005C2semidirect
2. #26041999C2semidirect
3. #26041984C2semidirect
4. #26041979C2semidirect
5. #26041964C2semidirect
6. Q8×D724C2semidirect
7. #26041936C2semidirect
8. #26041936C2semidirect
9. #26041894C2semidirect
10. #26041891C2semidirect
11. #26041811C2semidirect
12. #26041807C2semidirect
13. #26041755C2semidirect
14. C4×Q8×D181C2semidirect
15. #520×C181C2semidirect
176. C181#3516semidirect
190. #520D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260420052
#260419992
#260419842
#260419792
#260419642
Q8×D7242
#260419362
#260418942
#260418912
#260418112
#260418072
#260417552
C4×Q8×D1812
#520×C1812
#3516181